The mathematical framework used in physics and electrical engineering today to describe electromagnetic fields is based on work by Gauss, Faraday, and Ampere. The work is embodied in the following differential equations based on vector calculus, which today are referred to as Maxwell's equations:
            ∇              ·        D              =          ρ      ⁢                          ⁢              Gauss        '            ⁢      s      ⁢                          ⁢      law      ⁢                          ⁢      for      ⁢                          ⁢      electricity                  ∇              ·        B              =          0      ⁢                          ⁢              Gauss        '            ⁢                          ⁢      law      ⁢                          ⁢      for      ⁢                          ⁢      magnetism                  ∇              ×        E              =                  -                              ∂            B                                ∂            t                              ⁢                          ⁢              Faraday        '            ⁢      s      ⁢                          ⁢      law      ⁢                          ⁢      of      ⁢                          ⁢      induction                  ∇              ×        H              =          J      +                                    ∂            D                                ∂            t                          ⁢                                  ⁢                  Ampere          '                ⁢        s        ⁢                                  ⁢        law            
These equations were derived from experiments in the late 1800's with current-carrying conductors and are optimized to describe the electromagnetic effects from current-carrying conductors. These equations were derived under the assumption that only electromagnetic fields (E and B) are physical, and that the electromagnetic potentials φ (Electric Potential) and A (Magnetic Vector Potential), are purely mathematical constructs. These equations were thought to be complete at the time to describe all electromagnetic effects that could be observed from electrical conduction and convection currents.
By the 1980's the Aharonov-Bohm effect had proven the physicality (the reality) of the electromagnetic potentials, φ (Electric Potential) and A (Magnetic Vector Potential). The above equations by including only the fields and not their associated potentials end up not completely describing all the effects that are being observed from electrical convection currents.